
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39251695
70434
10.1038/s41598-024-70434-2
Article
Construction of improved comprehensive classes of estimators for population distribution function
Mustafa Manahil SidAhmed 1
Ahmad Sohaib 2
Aljohani Hassan M. 3
Alghamdi Fatimah M. 4
Aldallal Ramy 5
Elgarhy Mohammed 67
Almarzouki Sanaa Mohammed 8
Nasiru Suleman sulemanstat@gmail.com

9
1 https://ror.org/04yej8x59 grid.440760.1 0000 0004 0419 5685 Department of Statistics, Faculty of Science, University of Tabuk, Tabuk, Saudi Arabia
2 https://ror.org/04s9hft57 grid.412621.2 0000 0001 2215 1297 Department of Statistics, Quaid-I-Azam University Islamabad, Islamabad, Pakistan
3 https://ror.org/014g1a453 grid.412895.3 0000 0004 0419 5255 Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
4 https://ror.org/05b0cyh02 grid.449346.8 0000 0004 0501 7602 Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, 11671 Riyadh, Saudi Arabia
5 https://ror.org/04jt46d36 grid.449553.a 0000 0004 0441 5588 Department of Management, College of Business Administration in Hawtat Bani Tamim, Prince Sattam bin Abdulaziz University, Hawtat Bani Tamim, Saudi Arabia
6 https://ror.org/05pn4yv70 grid.411662.6 0000 0004 0412 4932 Department of Mathematics and Computer Science, Department of Basic Science, Faculty of Science, Beni Suef University, Beni-Suef, 62521 Egypt
7 Department of Basic Sciences, Higher Institute of Administrative Sciences, Belbeis, AlSharkia Egypt
8 grid.412125.1 0000 0001 0619 1117 Statistics Department, Faculty of Science, King Abdul Aziz University, Jeddah, Kingdom of Saudi Arabia
9 https://ror.org/00kpq4k75 Department of Statistics and Actuarial Science, School of Mathematical Sciences, C. K. Tedam University of Technology and Applied Sciences, Navrongo, Ghana
9 9 2024
9 9 2024
2024
14 209194 4 2024
16 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
The primary purpose of this article is to examine the issue of estimating the finite population distribution function from auxiliary information, such as population mean and rank of the auxiliary variables, that are already known. In order to better estimate the distribution function (DF) of a finite population, two improved estimators are developed. The bias and mean squared error of the suggested and existing estimators are derived up to the first order of approximation. To improve the efficiency of an estimators, we compare the suggested estimators with existing counterpart. Based on the numerical outcomes, it is to be noted that the suggested classes of estimators perform well using six actual data sets. The strength and generalization of the suggested estimators are also verified using a simulation analysis. Based on the result of actual data sets and a simulation study, we observe that the suggested estimator outperforms as compared to all existing estimators which is compared in this study.

Keywords

Distribution function (DF)
Simple random sampling
MSE
PRE
Simulation study
Visualization
Subject terms

Engineering
Mathematics and computing
http://dx.doi.org/10.13039/501100004242 Princess Nourah Bint Abdulrahman University PNURSP2024R735 Alghamdi Fatimah M. issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

In the literature on survey sampling, the use of auxiliary information progresses the precision of an estimators. The finest possible estimates of population metrics like mean, median, variance, standard deviation, etc. have previously been discovered by researchers. To achieve this goal, it is necessary to draw sample from the population; when the target population is uniform, a simple random sampling provide better result. When the study variable and the auxiliary variables have a high degree of association, then the rank of the auxiliary information is also associated to the study variable. The ratio and product estimators can enhance the accuracy of estimators when there is either a positive or negative association between the studied variable and the extra information. By consulting1–7, the researcher can further investigate these findings using auxiliary variables.

There is a substantial amount of literature available on the topic of population parameter estimate using different sampling approaches. But research based on distribution function (DF) has received less attention compared to the many estimators for estimating distinct finite population parameters under diverse sampling procedures in the literature. In order to determine what percentage of values are less than or equal to the threshold value, it is necessary to estimate a finite population DF. As an example, a doctor would wonder what percentage of the population get at least 20% of their caloric intake from cholesterol in their food. A soil scientist is interested in learning the poverty rate in a developing nation. Initially the technique for estimating the population DF was proposed by8. Some essential resources for learning how to estimate population DF using auxiliary information are given in9–16.

There is a substantial amount of literature available on the topic of population parameter estimate using different sampling approaches. But research based on distribution function (DF) has received less attention compared to the many estimators for estimating population parameters. In this paper we suggested improved classes of estimators for estimation of population DF using dual use of an auxiliary variables. Estimation of population DF is required when the percentage of particular values are less than or equal to the specific threshold. To check the robustness and generalizability we have utilized six real data sets and a simulation study.

The remaining of the article is designed as follows. In “Notation and symbols” section, the notations and symbols for the said work is given. The existing estimators were analyzed in “Existing estimators” section. In “Suggested estimators” section, we suggested two improved classes estimators for determining the DF. In “Numerical study” section, the empirical study are given. In “Simulation study” section, we also comportment a simulation study to test the efficacy of our proposed families of estimators using a simple random sample. In “Discussion” section, the numerical results are discussed. “Conclusion” section, provides conclusion of the article.

Notation and symbols

Let a population Ω={1,2,⋯,N} consist of N separate and identifiable units, we take a sample of size n from Ω using a SRSWOR. Let Y and X be the study variable and auxiliary variable. Consider Z is used for the rank of X. Let I(Y≤y) signify the indicator variable for Y, and I(X≤y) signify the display variable for X.F(y)=∑i=1NI(Yi≤y)/N,F^(y)=∑i=1nI(Yi≤y)/n,F(x)=∑i=1NI(Xi≤y)/N,

F^(x)=∑i=1nI(Xi≤y)/n, are the DF functions of Y and X for population and sample, respectively. Similarly,X¯=∑i=1NXi/N,X¯^=∑i=1nXi/n,Z¯=∑i=1NZi/N,Z¯^=∑i=1nZi/n.

ξ0=F^(y)-F(y)F(y),ξ1=F^(x)-F(x)F(x),ξ2=X¯^-X¯X¯andξ3=Z¯^-Z¯Z¯,

Eξ02=λCFy2,Eξ12=λCFx2,Eξ22=λCx2,Eξ32=λCrx2,Eξ0ξ1=λρ12CFyCFx,

E(ξ0ξ2)=λρ13CFyCx,E(ξ0ξ3)=λρ14CFyCrx,E(ξ1ξ2)=λρ23CFxCx,E(ξ1ξ3)=λρ24CFxCrx,

ρ12=∑i=1NI(Yi≤y)-F(y)2/(N-1),ρ22=∑i=1NI(Xi≤x)-F(x)2/(N-1),

ρ32=∑i=1N(Xi-X¯)2/(N-1),ρ42=∑i=1N(Zi-Z¯)2/(N-1),

CFy=ρ1/F(y),CFx=ρ2/F(x),Cx=ρ3/X¯,Crx=ρ4/Z¯,

ρ12=σ12/σ1σ2,ρ13=σ13/σ1σ3,ρ23=σ23/σ2σ3,ρ14=σ14/σ1σ4,ρ24=σ24/σ2σ4.

σ12=∑i=1N(I(Yi≤y)-F(y))(I(Xi≤x)-F(x))/(N-1),σ13=∑i=1N(I(Yi≤y)-F(y))(Xi-X¯)/(N-1),σ23=∑i=1N(I(Xi≤x)-F(x))(Xi-X¯)/(N-1),σ14=∑i=1N(I(Yi≤y)-F(y))(Zi-Z¯)/(N-1),σ24=∑i=1N(I(Xi≤x)-F(x))(Zi-Z¯)/(N-1),

where λ=(1/n-1/N).

let R1.232=ρ122+ρ132-2ρ12ρ13ρ23/1-ρ232. Similarly, R1.242=ρ122+ρ142-2ρ12ρ14ρ24/1-ρ242.

Existing estimators

Here, we take some adopted existing for population DF, which is given byThe usual estimator for DF, is given by:1 F^(y)=1n∑i=1nYi.

The variance of F^(y):2 Var(F^(y))=λF2(y)CFy2.

Reference17 give a ratio estimator for estimating F(y):3 F^R(Y)=F^(y)F(x)F^(x).

Bias(F^R(Y))≅λF(y)(CFx2-ρ12CFyCFx),

and4 MSE(F^R(Y))≅λF2(y)(CFy2+CFx2-2ρ12CFyCFx).

Reference18 suggested a product estimator for F(y):5 F^P(Y)=F^(y)F^(x)F(x).

Bias(F^P(Y))=λF(y)ρ12CFyCFx,

and6 MSE(F^P(Y))≅λF2(y)(CFy2+CFx2+2ρ12CFyCFx).

The regression estimator of F(y):7 F^Reg(Y)=F^(y)+w(F(x)-F^(x))

where w is constant.w(opt)=ρ12(ρY/ρX),

8 Varmin(F^Reg(Y))=λF2(y)CFy2(1-ρ122).

Reference19 suggested a difference estimator, given by:9 F^R,D(Y)=w1F^(y)+w2(F(x)-F^(x))

Bias(F^R,D(Y))=F(y)(w1-1)

andMSE(F^R,D(Y))≅F2(y)(w1-1)2+λF2(y)CFy2w12+λF2(x)CFx2w22

10 -2λF(y)F(x)ρ12CFyCFxw1w2.

wherew1opt=11+λCFy21-ρ122,

w2(opt)=F(y)ρ12CFyF(x)CFx{1+λCFy2(1-ρ122)},

Using w1opt, w1opt we got:11 MSEmin(F^R,D(Y))≅λF2(y)CFy2(1-ρ122)1+λCFy2(1-ρ122).

Reference20 suggested exponential type estimators, given by:12 F^BT,R(Y)=F^(y)expF(x)-F^(x)F^(x)+F(x),

13 F^BT,P(Y)=F^(y)expF^(x)-F(x)F^(x)+F(x).

Bias(F^BT,R(Y))≅λF(y)3CFx28-ρ12CFyCFx2,

14 MSE(F^BT,R(Y))≅λF(y)24(4CFy2+CFx2-4ρ12CFyCFx),

andBias(F^BT,P(Y))≅λF(y)ρ12CFyCFx2-CFx28,

15 MSE(F^BT,P(Y))≅λF2(y)4(4CFy2+CFx2+4ρ12CFyCFx).

Reference21 suggested the following estimator, given by:16 F^S(Y)=F^(y)expα(F(x)-F^(x))α(F(x)+F^(x))+2β

The estimator F^S(Y) reduces to F^BT,R(Y) and F^BT,P(Y) when (α=1,β=0) and (α=-1,β=0), respectively.Bias(F^S(Y))≅λF(y)3Θ2CFx28-Θρ12CFyCFx2,

and17 MSE(F^S(Y))≅λF2(y)4(4CFy2+Θ2CFx2-4Θρ12CFyCFx),

where Θ=αF(x)/(αF(x)+β).

Reference22 suggested a generalized ratio-type exponential estimator, given by:18 F^GK(Y)=w3F^(y)+w4(F(x)-F^(x))expα(F(x)-F^(x))α(F(x)+F^(x))+2β,

Bias(F^GK(Y))≅F(y)-w3F(y)+38w3Θ2F(y)λCFy2+12w4ΘF(x)λCFx2

-12w3ΘF(y)λρ12CFyCFx,

andMSE(F^GK(Y))≅F2(y)(w3-1)2+w32F2(y)λCFy2+w42F2(x)λCFx2+Θ2F2(y)λCFx2w32+2w3w4ΘF(y)F(x)λCFx2-34w3Θ2F2(y)λCFx2-w4ΘF(y)F(x)λCFx2+w3ΘF2(y)λρ12CFyCFx-2w32ΘF2(y)λρ12CFyCFx-2w3w4F(y)F(x)λρ12CFyCFx.

w3(opt)=8-λΘ2CFx28{1+λCFy2(1-ρ122)}

w4(opt)=F(y)λΘ3CFx3+8ρ12CFy-λΘ2ρ12CFyCFx2-4ΘCFx{1-λCFy2(1-ρ122)}8F(x)CFx{1+λCFy2(1-ρ122)},

19 MSEminF^GK(Y)≅λF2(y){64CFy2(1-ρ122)-λΘ4CFx4-16λΘ2CFy2CFx2(1-ρ122)}64{1+λCFy2(1-ρ122)}.

Here, (19) may be written as20 MSEminF^GK(Y)≅Varmin(F^st∗Reg(Y))-λ2F2yΘ2CFx2+8CFy21-ρ122264{1+λCFy2(1-ρ122)},

Suggested estimators

By incorporating the auxiliary variables, the design and estimation stages of an estimator can take benefit. When the study variable is associated with the auxiliary variable, then rank of the auxiliary variable is also correlated with each other. Therefore, the rank of the auxiliary variable can be considered as an additional information, it helps to improve the estimator accuracy. To calculate an approximation of the population distribution function, we use more information regarding the sample means and the rank of the auxiliary variable, along with the sample distribution functions of F(y) and F(x).

First improved class of estimator

Taking motivation from F^R,D(y), F^S(y) and average of F^BT,R(y) and F^BT,P(y), our first proposed class of the estimator, is given by:F^Pr1Y=12F^(y)expF(x)-F^(x)F^(x)+F(x)+expF^(x)-F(x)F^(x)+F(x)+w5F(x)-F^(x)+w6F^(y)+w7X¯-X¯^expαF(x)-F^(x)α(F(x)+F^(x))+2β.

The estimator F^Pr1(Y), is expressed as:21 F^Pr1Y=Fy1+ξ01+w6-w5ξ1-w7ξ2+18Θ2Fyξ121-12Θξ1+38Θ2ξ12+⋯.

By simplifying (21), we have22 F^Pr1(Y)-F(y)≅w6F(y)+F(y)ξ0+w6F(y)ξ0-12ΘF(y)ξ1+Θ2F(y)ξ12-12ΘF(y)ξ0ξ1-w5ξ1+12Θw5ξ12-12Θw6F(y)ξ1+38Θ2w6F(y)ξ12-12Θw6F(y)ξ0ξ1-w7ξ2+12Θw7ξ1ξ2

The bias and MSE of F^Pr1(Y), are given asBias(F^Pr1(Y))≅12Θ2F(y)λCFx2-12ΘF(y)λρ12CFyCFx+12w5ΘλCFx2+w6F(y)+38w6Θ2F(y)λCFx2-12w6ΘF(y)λρ12CFyCFx+12w7Θλρ23CFxCx,

and 23 MSE(F^Pr1(Y))≅-ΘF2(y)λρ12CFyCFx+32w6Θ2F2(y)λCFx2+w62Θ2F2(y)λCFx2+w5ΘF(y)λCFx2-2w62ΘF2(y)λρ12CFyCFx+F2(y)λCFy2+w62F2(y)+w7ΘF(y)λρ23CFxCx-3w6ΘF2(y)λρ12CFyCFx-2w5F(y)λρ12CFyCFx-2w7F(y)λρ13CFyCx-2w6w7F(y)λρ13CFyCx+14Θ2F2(y)λCFx2+2w6F2(y)λCFy2+w52λCFx2+2w5w6ΘF(y)λCFx2-2w5w6F(y)λρ12CFyCFx+2w5w7λρ23CFxCx+w62F2yλCFy2+m72λCx2+2w6w7ΘFyλρ23CFxCx.

The optimum values for w5, w6 and w7, determined (23) are given as:w5(opt)=F(y)λΘ3CFx3ρ232-λΘ2CFyCFx2ρ13ρ23-4λΘCFy2CFxρ12ρ13ρ23-λΘ3CFx3+λΘ2CFyCFx2ρ12+2λΘCFy2CFxρ122+2λΘCFy2CFxρ132+2λΘCFy2CFxρ232-2λΘCFy2CFx-2ΘCFxρ232+4CFyρ13ρ23+2ΘCFx-4CFyρ124CFx-2λCFy2ρ12ρ13ρ23+λCFy2ρ122+λCFy2ρ132+λCFy2ρ232-λCFy2+ρ232-1

w6(opt)=-(Θ2CFx2ρ232-8CFy2ρ12ρ13ρ23-(Θ2CFx2+4CFy2ρ122+4CFy2ρ132+4CFy2ρ232-4CFy2)λ4(-2λCFy2ρ12ρ13ρ23+λCFy2ρ122+λCFy2ρ132+λCFy2ρ232-λCFy2+ρ232-1)

andw7(opt)=-F(y)CFy(λΘ2CFx2ρ12ρ23-λΘ2CFx2ρ13-4ρ12ρ23+4ρ13)4Cx(-2λCFy2ρ12ρ13ρ23+λCFy2ρ122+λCFy2ρ132+λCFy2ρ232-λCFy2+ρ232-1),

24 MSEminF^Pr1(Y)≅F2(y)λ{16CFy2(1-R1.232)-λΘ4CFx4-8λΘ2CFx2CFy2(1-R1.232)}16{1+λCFy2(1-R1.232)}

whereR1.232=ρ122+ρ132-2ρ12ρ13ρ23/1-ρ232.

Second modified class of estimator

Both the design and estimating stages of an estimator can benefit by incorporating of additional information. When the study variable is highly connected with the auxiliary variable, the rank of the auxiliary variable will also be connected with the study variable. That's why the rank of the auxiliary variable can serve as yet another piece of supplementary data. Using the idea of rank, we proposed a second modified class of estimator, given by:F^Pr2Y=12F^(y)expF(x)-F^(x)F^(x)+F(x)+expF^(x)-F(x)F^(x)+F(x)+w8F(x)-F^(x)+w9F^(y)+w10Z¯-Z¯^expα(F(x)-F^(x))α(F(x)+F^(x))+2β.

The estimator F^Pr2(Y), can also be written as25 F^Pr2Y=Fy1+ξ01+w9-w8ξ1-w10ξ3+18Θ2Fyξ121-12Θξ1+38Θ2ξ12+⋯.

26 F^Pr2Y-F(y)≅w9Fy+Fyξ0+w9Fyξ0-12ΘFyξ1+12Θ2Fyξ12-12ΘFyξ0ξ1-w8ξ1+12Θw8ξ12-12Θw9Fyξ1+38Θ2w9Fyξ12-12Θw9Fyξ0ξ1-w10ξ3+12Θw10ξ1ξ3.

Bias(F^Pr2(Y))≅12Θ2F(y)λCFx2-12ΘF(y)λρ12CFyCFx+12w8ΘλCFx2+w9F(y)+38w9Θ2F(y)λCFx2-12w9ΘF(y)λρ12CFyCFx+12w10Θλρ24CFxCrx,

27 MSE(F^Pr2(Y)≅-ΘF2(y)λρ12CFyCFx+32w9Θ2F2(y)λCFx2+w92Θ2F2(y)λCFx2+w8ΘF(y)λCFx2-2w92ΘF2(y)λρ12CFyCFx+F2(y)λCFy2+w92F2(y)+w10ΘF(y)λρ24CFxCrx-3w9ΘF2(y)λρ12CFyCFx-2w8F(y)λρ12CFyCFx-2w10F(y)λρ14CFyCrx-2w9w10F(y)λρ14CFyCrx+14Θ2F2(y)λCFx2+2w9F2(y)λCFy2+w82λCFx2+2w8w9ΘF(y)λCFx2-2w8w9F(y)λρ12CFyCFx+2w8w10λρ24CFxCrx+w92F2(y)λCFy2+m72λCrx2+2w9w10ΘF(y)λρ24CFxCrx.

The values of w8, w9 and w10, are given by:w8(opt)=F(y)λΘ3CFx3ρ242-λΘ2CFyCFx2ρ14ρ24-4λΘCFy2CFxρ12ρ14ρ24-λΘ3CFx3+λΘ2CFyCFx2ρ12+2λΘCFy2CFxρ122+2λΘCFy2CFxρ142+2λΘCFy2CFxρ242-2λΘCFy2CFx-2ΘCFxρ242+4CFyρ14ρ24+2ΘCFx-4CFyρ124CFx-2λCFy2ρ12ρ14ρ24+λCFy2ρ122+λCFy2ρ142+λCFy2ρ242-λCFy2+ρ242-1

w9(opt)=-(Θ2CFx2ρ242-8CFy2ρ12ρ14ρ24-(Θ2CFx2+4CFy2ρ122+4CFy2ρ142+4CFy2ρ242-4CFy2)λ4-2λCFy2ρ12ρ14ρ24+λCFy2ρ122+λCFy2ρ142+λCFy2ρ242-λCFy2+ρ242-1,

andw10(opt)=-F(y)CFy(λΘ2CFx2ρ12ρ24-λΘ2CFx2ρ14-4ρ12ρ24+4ρ14)4Crx(-2λCFy2ρ12ρ14ρ24+λCFy2ρ122+λCFy2ρ142+λCFy2ρ242-λCFy2+ρ242-1),

The MSE of F^Pr2(Y) at the values of w8, w9, and w10, is given by:28 MSEminF^Pr2(Y)≅F2(y)λ{16CFy2(1-R1.242)-λΘ4CFx4-8λΘ2CFx2CFy2(1-R1.242)}16{1+λCFy2(1-R1.242)}

where R1.242=ρ122+ρ142-2ρ12ρ14ρ24/1-ρ242 (Table 1). Table 1 Some elements of existing and suggested estimators.

α	β	F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	
1	CFx	F^GK(1)(Y)	F^Pr1(1)(Y)	F^Pr2(1)(Y)	
1	β2	F^GK(2)(Y)	F^Pr1(2)(Y)	F^Pr2(2)(Y)	
β2	CFx	F^GK(3)(Y)	F^Pr1(3)(Y)	F^Pr2(3)(Y)	
CFx	β2	F^GK(4)(Y)	F^Pr1(4)(Y)	F^Pr2(4)(Y)	
1	ρ12	F^GK(5)(Y)	F^Pr1(5)(Y)	F^Pr2(5)(Y)	
CFx	ρ12	F^GK(6)(Y)	F^Pr1(6)(Y)	F^Pr2(6)(Y)	
ρ12	CFx	F^GK(7)(Y)	F^Pr1(7)(Y)	F^Pr2(7)(Y)	
β2	ρ12	F^GK(8)(Y)	F^Pr1(8)(Y)	F^Pr2(8)(Y)	
ρ12	β2	F^GK(9)(Y)	F^Pr1(9)(Y)	F^Pr2(9)(Y)	
1	NF(x)	F^GK(10)(Y)	F^Pr1(10)(Y)	F^Pr2(10)(Y)	

Numerical study

We take a numerical analysis to compare the existing and the suggested classes of estimators. Six actual data sets are used for this purpose. Tables 2, 3, 4, 5, 6 and 7 present aggregate statistics for the provided data. PRE of an estimator F^i(Y) concerning F^(y) is Table 2 Data description using Population 1.

Parameter	Value	Parameter	x and y (value)	
X¯,Y¯	Q1(x),Q1(y)	X~,Y~	Q3(x),Q3(y)	
N	923	F(y)	0.7703	0.2556	0.5016	0.7497	
n	180	CFy	0.5463	1.7070	0.9973	0.5780	
λ	0.00447	F(x)	0.7693	0.2503	0.5005	0.7508	
X¯	11,440.5	CFx	0.5480	1.7317	0.9995	0.5764	
Cx	1.86453	ρ12	0.8930	0.8711	0.8462	0.8930	
Z¯	461.000	ρ13	−0.6640	−0.2861	−0.4416	−0.6465	
Crx	0.57703	ρ23	−0.6753	−0.2838	0.4480	−0.6563	
		ρ14	−0.7169	−0.7424	−0.8286	−0.7402	
		ρ24	−0.7298	−0.7503	−0.8660	−0.7492	
		β2	1.6333	1.3295	1.0000	1.3449	

Table 3 Data description using Population 2.

Parameter	Value	Parameter	x and y (value)	
X¯,Y¯	Q1(x),Q1(y)	X~,Y~	Q3(x),Q3(y)	
N	923	F(y)	0.7703	0.2556	0.5016	0.7497	
n	180	CFy	0.5463	1.7070	0.9973	0.5780	
λ	0.00447	F(x)	0.7291	0.2524	0.5016	0.7508	
X¯	333.165	CFx	0.6098	1.7218	0.9973	0.5764	
Cx	1.32809	ρ12	0.8727	0.1910	0.8917	0.9162	
Z¯	461.000	ρ13	−0.7385	−0.3656	−0.5366	−0.7221	
Crx	0.57703	ρ23	−0.7120	−0.1046	−0.5419	−0.7298	
		ρ14	−0.7223	−0.7424	−0.8486	−0.7430	
		ρ24	−0.7697	−0.7503	−0.8660	−0.7491	
		β2	1.0634	1.2990	1.0000	1.3449	

Table 4 Data description using Population 3.

Parameter	Value	Parameter	x and y (value)	
X¯,Y¯	Q1(x),Q1(y)	X~,Y~	Q3(x),Q3(y)	
N	69	F(y)	0.7246	0.2464	0.5072	0.7536	
n	10	CFy	0.6209	1.7618	0.9928	0.5759	
λ	0.08550	F(x)	0.7681	0.2464	0.5072	0.7536	
X¯	4954.44	CFx	0.5535	1.7618	0.9928	0.5759	
Cx	1.42478	ρ12	0.6607	0.7658	0.9420	0.7658	
Z¯	35.0000	ρ13	−0.7129	−0.3612	−0.5709	−0.7424	
Crx	0.57321	ρ23	−0.7745	−0.3650	−0.5427	−0.7584	
		ρ14	−0.7168	−0.7109	−0.8558	−0.7008	
		ρ24	−0.7310	−0.7464	−0.8660	−0.7464	
		β2	1.6144	1.3857	1.0000	1.3857	

Table 5 Data description using Population 4.

Parameter	Value	Parameter	x and y (value)	
X¯,Y¯	Q1(x),Q1(y)	X~,Y~	Q3(x),Q3(y)	
N	69	F(y)	0.7246	0.2464	0.5072	0.7536	
n	10	CFy	0.6209	1.7618	0.9928	0.5759	
λ	0.08550	F(x)	0.7391	0.2464	0.5072	0.7536	
X¯	4591.72	CFx	0.5984	1.7618	0.9928	0.5759	
Cx	1.37554	ρ12	0.8159	0.6878	0.7100	0.7658	
Z¯	35.0000	ρ13	−0.7102	−0.3677	−0.5519	−0.7354	
Crx	0.57321	ρ23	−0.7556	−0.3717	−0.7689	−0.7584	
		ρ14	−0.7282	−0.7424	−0.7903	−0.7008	
		ρ24	−0.7606	−0.7464	−0.8660	−0.7464	
		β2	1.1863	1.385	1.0000	1.3857	

Table 6 Data description using Population 5.

Parameter	Value	Parameter	x and y (value)	
X¯,Y¯	Q1(x),Q1(y)	X~,Y~	Q3(x),Q3(y)	
N	50	F(y)	0.6800	0.2400	0.5000	0.7600	
n	5	CFy	0.6929	1.7975	1.0102	0.5677	
λ	0.18000	F(x)	0.5800	0.2600	0.5000	0.7600	
X¯	78.2900	CFx	0.8596	1.7042	1.0102	0.5677	
Cx	0.27229	ρ12	−0.1494	−0.0128	−0.120	−0.2061	
Z¯	25.5000	ρ13	0.2841	0.3322	0.2292	0.2174	
Crx	0.57159	ρ23	−0.8094	−0.6202	−0.7894	−0.8215	
		ρ14	0.2526	0.2402	0.1843	0.1882	
		ρ24	−0.8551	−0.7599	−0.8663	−0.7399	
		β2	1.1050	1.1975	1.0000	1.4824	

Table 7 Data description using Population 6.

Parameter	Value	Parameter	x and y (value)	
X¯,Y¯	Q1(x),Q1(y)	X~,Y~	Q3(x),Q3(y)	
N	854	F(y)	0.8934	0.2494	0.5012	0.7506	
n	140	CFy	0.3456	1.7358	0.9982	0.5768	
λ	0.18000	F(x)	0.8279	0.2494	0.5012	0.7506	
X¯	37,600.1	CFx	0.4563	1.7358	0.9982	0.5768	
Cx	3.85089	ρ12	0.6870	0.7623	0.7658	0.7498	
Z¯	427.500	ρ13	−0.4550	−0.1413	−0.2260	−0.3351	
Crx	0.57700	ρ23	−0.4118	−0.1448	−0.2345	−0.3522	
		ρ14	−0.5137	−0.6914	−0.7834	−0.7003	
		ρ24	−0.6538	−0.7494	−0.8660	−0.7490	
		β2	1.1050	1.1975	1.0000	1.4824	

PREF^i(Y),F^(y)=VarF^(y)MSEF^i(Y)×100,

Population-I: [Source:23]:

Y: Number of instructors,

X: number of pupils,

Z: order of X.

Population-II: [Source:23].

Y: number of an instructors,

X: number of classes,

Z: order of X.

Population III: [Source:24].

Y: the number of fish caught in 1995,

X: The number of fish caught in 1994,

Z: order of X.

Population IV: [Source:24]:

Y: the number of fish caught in 1995,

X: The number of fish caught in 1993,

Z: order of X.

Population V: [Source:25].

Y: The eggs formed in 1990,

X: The amount of per dozen eggs in 1990,

Z: order of X.

Population VI: [Source:26].

Y: The production of apple in 1999,

X: The number of apple plants in 1999,

Z: order of X.

Simulation study

We have generated three populations of size 1000 from a multivariate normal distribution with different covariance matrices. All the populations have different correlations i.e., the auxiliary variable (X) and study variable (Y) are negatively correlated in Population I, but the same variables are positively correlated in Population II, and strongly positive association in case of Population III correlation.

Population-I:μ1=55

and∑1=4-9.0-9.064

ρXY=-0.590220

Population-II: μ2=55,

∑2=49.59.563

ρXY=0.612254

Population-III:μ3=55,

∑3=24610

ρXY= 0.902645.

The Percentage Relative Efficiency (PRE) is calculated as follows:PREF^i(Y),F^(y)=VarF^(y)MSEF^i(Y)×100,

The results of MSE and PRE are given in Tables 16 and 17. Here we can only point out the best results of MSEs and PREs in these tables when Θ=αF(x)αF(x)+β if α=CFxandβ=β2.

Discussion

Table 1, include some elements of the existing and suggested classes of estimators. From the numerical results, which are presented in Tables 8, 9, 10, 11, 12, 13, 14 and 15, We want to bring back the fact that PRE varies for the different choices of a and b. For the data sets, if we use (α=1 and β=ρ12), (α=CFx and β=ρ12) and (α=β2 and β=ρ12) we get the largest values of PRE of all families among different classes. Consequently, the ideal results from the families of estimators are attained by choosing and as the coefficients of variation, kurtosis, and correlation, respectively. It is also found that the second proposed class of estimators F^Pr2(Y) behaves slightly better than the first proposed family of estimators F^Pr1(Y), shown in Tables 8, 9, 10, 11, 12, 13, 14 and 15 which demonstrate the average gain inadequacies for the six populations, respectively, while the first suggested class of estimator F^Pr1(Y) performs better over the second suggetsed class of estimator F^Pr2(Y) with substantial normal improvement inadequacies for the second population. While from Tables 8, 9, 10, 11, 12, 13, 14 and 15 the PRE of all families are diminishing diagonally the values of (α=1 and β=NF(x)). Table 8 Percentage relative efficiency using Population 1, 2, 3 when {x=X¯,y=Y¯}.

Estimators	Population 1	Population 2	Population 3	
F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	
F^GK1Y	F^Pr1(1)(Y)	F^Pr2(1)(Y)	493.99	511.31	517.30	419.78	475.32	431.20	181.17	220.67	228.54	
F^GK(2)(Y)	F^Pr1(2)(Y)	F^Pr2(2)(Y)	493.94	511.19	517.18	419.74	475.20	431.10	180.82	219.78	227.61	
F^GK(3)(Y)	F^Pr1(3)(Y)	F^Pr2(3)(Y)	494.05	511.47	517.46	419.83	475.45	431.31	181.53	221.65	229.55	
F^GK(4)(Y)	F^Pr1(4)(Y)	F^Pr2(4)(Y)	493.93	511.17	517.17	419.73	475.19	431.09	180.78	219.68	227.51	
F^GK(5)(Y)	F^Pr1(5)(Y)	F^Pr2(5)(Y)	493.97	511.25	517.25	419.77	475.27	431.16	181.11	220.51	228.37	
F^GK(6)(Y)	F^Pr1(6)(Y)	F^Pr2(6)(Y)	493.94	511.20	517.19	419.75	475.23	431.12	180.95	220.08	227.93	
F^GK(7)(Y)	F^Pr1(7)(Y)	F^Pr2(7)(Y)	493.98	511.29	517.29	419.78	475.30	431.18	181.04	220.32	228.17	
F^GK(8)(Y)	F^Pr1(8)(Y)	F^Pr2(8)(Y)	494.02	511.38	517.38	419.80	475.38	431.26	181.47	221.47	229.37	
F^GK(9)(Y)	F^Pr1(9)(Y)	F^Pr2(9)(Y)	493.93	511.18	517.18	419.74	475.20	431.09	180.79	219.70	227.53	
F^GK(10)(Y)	F^Pr1(10)(Y)	F^Pr2(10)(Y)	493.93	511.17	517.16	419.73	475.18	431.08	180.76	219.63	227.46	
	F^(y)			100.00			100.00				100.00	
	F^R(Y)			465.57			336.04				161.16	
	F^P(Y)			25.470			23.310				33.340	
	F^BT,R(Y)			281.07			296.44				164.00	
	F^BT,P(Y)			46.570			43.750				55.940	
	F^Reg(Y)			493.79			419.59				177.46	
	F^R,D(Y)			493.93			419.73				180.76	

Table 9 Percentage relative efficiency using Population 1, 2, 3 when x=Q1(x),y=Q1(y).

Estimators	Population 1	Population 2	Population 3	
F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	
F^GK(1)(Y)	F^Pr1(1)(Y)	F^Pr2(1)(Y)	415.95	418.83	449.59	105.10	119.99	227.16	268.65	273.50	297.65	
F^GK(2)(Y)	F^Pr1(2)(Y)	F^Pr2(2)(Y)	415.94	418.80	449.57	105.09	119.98	227.15	268.54	273.28	297.40	
F^GK(3)(Y)	F^Pr1(3)(Y)	F^Pr2(3)(Y)	416.02	418.96	449.74	105.12	120.03	227.23	269.51	275.29	299.59	
F^GK(4)(Y)	F^Pr1(4)(Y)	F^Pr2(4)(Y)	415.96	418.85	449.62	105.10	119.99	227.17	268.60	273.83	298.00	
F^GK(5)(Y)	F^Pr1(5)(Y)	F^Pr2(5)(Y)	416.00	418.92	449.69	105.20	120.24	227.67	269.45	275.16	299.46	
F^GK(6)(Y)	F^Pr1(6)(Y)	F^Pr2(6)(Y)	416.09	419.10	449.89	105.27	120.39	227.96	270.78	277.99	301.54	
F^GK(7)(Y)	F^Pr1(7)(Y)	F^Pr2(7)(Y)	415.95	418.82	449.58	105.09	119.97	227.13	268.55	273.3)	297.43	
F^GK(8)(Y)	F^Pr1(8)(Y)	F^Pr2(8)(Y)	416.16	419.27	450.07	105.29	120.46	228.13	271.88	280.38	305.18	
F^GK(9)(Y)	F^Pr1(9)(Y)	F^Pr2(9)(Y)	415.94	418.80	449.57	105.93	119.97	227.13	268.48	273.15	297.27	
F^GK(10)(Y)	F^Pr1(10)(Y)	F^Pr2(10)(Y)	415.93	418.78	449.54	105.09	119.97	227.13	268.39	271.97	297.06	
	F^(y)			100.00			100.00				100.00	
	F^R(Y)			381.08			61.280				213.53	
	F^P(Y)			18.950			25.700				19.220	
	F^BT,R(Y)			267.67			94.190				206.54	
	F^BT,P(Y)			46.700			69.100				49.600	
	F^Reg(Y)			414.63			103.79				241.84	
	F^R,D(Y)			415.93			105.09				268.39	

Table 10 Percentage relative efficiency using Population 1, 2, 3 when x=X~,y=Y~.

Estimators	Population 1	Population 2	Population 3	
F^GK(1)(Y)	F^Pr1(1)(Y)	F^Pr2(Y)	F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	
F^GK(1)(Y)	F^Pr1(1)(Y)	F^Pr2(1)(Y)	351.58	358.79	404.92	488.43	498.33	551.21	140.84	898.79	913.16	
F^GK(2)(Y)	F^Pr1(2)(Y)	F^Pr2(2)(Y)	351.58	358.79	404.92	488.43	498.33	551.21	140.56	898.77	913.10	
F^GK(3)(Y)	F^Pr1(3)(Y)	F^Pr2(3)(Y)	351.78	358.79	404.92	488.43	498.33	551.21	140.86	898.79	913.17	
F^GK(4)(Y)	F^Pr1(4)(Y)	F^Pr2(4)(Y)	351.58	358.79	404.92	488.43	498.33	551.21	140.32	898.75	913.05	
F^GK(5)(Y)	F^Pr1(5)(Y)	F^Pr2(5)(Y)	351.59	358.81	404.94	488.45	498.35	551.24	141.67	898.97	913.56	
F^GK(6)(Y)	F^Pr1(6)(Y)	F^Pr2(6)(Y)	351.59	358.81	404.94	488.44	498.35	551.23	141.41	898.94	913.50	
F^GK(7)(Y)	F^Pr1(7)(Y)	F^Pr2(7)(Y)	351.57	358.77	404.89	488.43	498.30	551.19	138.82	898.60	911.74	
F^GK(8)(Y)	F^Pr1(8)(Y)	F^Pr2(8)(Y)	351.59	358.81	404.94	488.45	498.35	551.24	141.70	898.98	913.57	
F^GK(9)(Y)	F^Pr1(9)(Y)	F^Pr2(9)(Y)	351.57	358.77	404.89	488.43	498.30	551.19	138.56	989.59	911.69	
F^GK(10)(Y)	F^Pr1(10)(Y)	F^Pr2(10)(Y)	351.53	358.69	404.80	488.37	489.19	551.07	101.36	896.49	908.20	
	F^(y)			100.00			100.00				100.00	
	F^R(Y)			324.29			461.49				861.32	
	F^P(Y)			27.060			26.450				25.790	
	F^BT,R(Y)			248.08			279.06				324.69	
	F^BT,P(Y)			47.640			46.690				45.620	
	F^Reg(Y)			351.08			487.23				888.06	
	F^R,D(Y)			351.53			488.37				896.49	

Table 11 Percentage relative efficiency using Population 1, 2, 3 when x=Q3(x),y=Q3(y).

Estimators	Population 1	Population 2	Population 3	
F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	
F^GK(1)(Y)	F^Pr1(1)(Y)	F^Pr2(1)(Y)	494.04	510.24	523.86	621.69	647.49	651.53	245.26	288.33	270.29	
F^GK(2)(Y)	F^Pr1(2)(Y)	F^Pr2(2)(Y)	493.99	510.12	523.81	621.62	647.33	651.36	244.78	287.10	269.15	
F^GK(3)(Y)	F^Pr1(3)(Y)	F^Pr2(3)(Y)	494.10	510.39	524.09	621.77	647.70	651.74	245.76	289.70	271.56	
F^GK(4)(Y)	F^Pr1(4)(Y)	F^Pr2(4)(Y)	493.98	510.10	523.79	621.61	647.32	651.35	244.72	286.97	269.02	
F^GK(5)(Y)	F^Pr1(5)(Y)	F^Pr2(5)(Y)	494.02	210.18	523.88	621.66	647.42	651.45	245.12	287.96	269.95	
F^GK(6)(Y)	F^Pr1(6)(Y)	F^Pr2(6)(Y)	493.99	510.14	523.83	621.63	647.36	651.39	244.91	287.43	269.45	
F^GK(7)(Y)	F^Pr1(7)(Y)	F^Pr2(7)(Y)	494.04	510.22	523.92	621.68	647.48	651.51	245.13	287.98	269.97	
F^GK(8)(Y)	F^Pr1(8)(Y)	F^Pr2(8)(Y)	494.07	510.30	524.00	621.72	647.58	651.62	245.59	289.24	271.13	
F^GK(9)(Y)	F^Pr1(9)(Y)	F^Pr2(9)(Y)	493.99	510.11	523.80	621.62	647.33	651.36	244.74	287.03	269.08	
F^GK(10)(Y)	F^Pr1(10)(Y)	F^Pr2(10)(Y)	493.98	510.09	523.79	621.6	647.30	651.33	244.68	286.87	268.93	
	F^(y)			100.00			100.00				100.00	
	F^R(Y)			468.75			598.06				213.53	
	F^P(Y)			26.04			25.730				27.780	
	F^BT,R(Y)			279.25			298.47				206.54	
	F^BT,P(Y)			46.750			46.250				49.600	
	F^Reg(Y)			493.83			621.46				241.84	
	F^R,D(Y)			493.98			621.60				244.68	

Table 12 Percentage relative efficiency using Population 4, 5, 6 when {x=X¯,y=Y¯}.

Estimators	Population 4	Population 5	Population 6	
F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	
F^GK(1)(Y)	F^Pr1(1)(Y)	F^Pr2(1)(Y)	303.27	323.67	330.99	111.54	121.05	118.69	189.48	203.23	191.16	
F^GK(2)(Y)	F^Pr1(2)(Y)	F^Pr2(2)(Y)	301.67	321.31	329.60	111.37	120.67	118.33	189.46	203.17	191.10	
F^GK(3)(Y)	F^Pr1(3)(Y)	F^Pr2(3)(Y)	303.87	325.16	331.53	111.62	121.22	118.87	189.51	203.30	191.22	
F^GK(4)(Y)	F^Pr1(4)(Y)	F^Pr2(4)(Y)	301.58	321.13	329.42	111.29	120.49	118.15	189.45	203.17	191.10	
F^GK(5)(Y)	F^Pr1(5)(Y)	F^Pr2(5)(Y)	303.06	323.19	330.51	119.42	143.76	140.81	189.47	203.21	191.14	
F^GK(6)(Y)	F^Pr1(6)(Y)	F^Pr2(6)(Y)	301.81	321.63	329.93	120.76	148.63	145.52	189.46	203.19	191.12	
F^GK(7)(Y)	F^Pr1(7)(Y)	F^Pr2(7)(Y)	303.13	323.35	330.67	110.97	119.80	117.47	189.47	203.21	191.14	
F^GK(8)(Y)	F^Pr1(8)(Y)	F^Pr2(8)(Y)	303.62	324.54	331.89	118.76	141.50	138.62	189.50	203.28	191.21	
F^GK(9)(Y)	F^Pr1(9)(Y)	F^Pr2(9)(Y)	301.63	321.22	329.51	110.95	119.75	117.63	189.45	203.17	191.10	
F^GK(10)(Y)	F^Pr1(10)(Y)	F^Pr2(10)(Y)	301.52	321.99	329.74	110.93	119.70	117.38	189.45	203.17	191.10	
	F^(y)			100.00			100.00				100.00	
	F^R(Y)			280.87			34.370				17.380	
	F^P(Y)			28.490			51.640				107.62	
	F^BT,R(Y)			224.30			63.690				189.38	
	F^BT,P(Y)			49.400			83.380				189.45	
	F^Reg(Y)			299.22			101.28				189.13	
	F^R,D(Y)			301.52			110.93				41.680	

Table 13 Percentage relative efficiency using Population 4, 5, 6 when x=Q1(x),y=Q1(y).

Estimators	Population 4	Population 5	Population 6	
F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	
F^GK(1)(Y)	F^Pr1(1)(Y)	F^Pr2(1)(Y)	216.53	221.15	251.90	158.54	179.63	173.36	240.55	241.12	260.94	
F^GK(2)(Y)	F^Pr1(2)(Y)	F^Pr2(2)(Y)	216.44	221.97	251.69	158.41	179.33	173.07	240.54	241.11	260.93	
F^GK(3)(Y)	F^Pr1(3)(Y)	F^Pr2(3)(Y)	217.22	223.59	253.54	159.50	181.82	175.48	240.60	241.23	261.06	
F^GK(4)(Y)	F^Pr1(4)(Y)	F^Pr2(4)(Y)	216.65	221.41	251.19	158.76	180.13	173.85	240.56	241.14	260.96	
F^GK(5)(Y)	F^Pr1(5)(Y)	F^Pr2(5)(Y)	217.32	223.81	253.78	186.88	267.89	257.67	240.60	241.22	261.05	
F^GK(6)(Y)	F^Pr1(6)(Y)	F^Pr2(6)(Y)	218.52	226.37	256.71	185.40	261.80	251.90	240.68	241.39	261.23	
F^GK(7)(Y)	F^Pr1(7)(Y)	F^Pr2(7)(Y)	216.42	221.94	251.65	158.18	178.80	171.56	240.54	241.11	260.93	
F^GK(8)(Y)	F^Pr1(8)(Y)	F^Pr2(8)(Y)	219.48	228.48	259.13	184.96	260.00	250.20	240.74	241.52	261.37	
F^GK(9)(Y)	F^Pr1(9)(Y)	F^Pr2(9)(Y)	216.37	221.84	251.54	158.18	178.80	171.56	240.54	241.10	260.92	
F^GK(10)(Y)	F^Pr1(10)(Y)	F^Pr2(10)(Y)	216.31	221.72	251.40	158.19	178.80	171.58	240.53	241.09	260.91	
	F^(y)			100.00			100.00				100.00	
	F^R(Y)			160.14			51.000				19.560	
	F^P(Y)			19.810			26.780				210.36	
	F^BT,R(Y)			177.87			80.850				238.73	
	F^BT,P(Y)			51.60			81.470				240.53	
	F^Reg(Y)			189.77			100.02				205.05	
	F^R,D(Y)			216.30			158.18				49.690	

Table 14 Percentage relative efficiency using Population 4, 5, 6 when x=X~,y=Y~.

Estimators	Population 4	Population 5	Population 6	
F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	
F^GK(1)(Y)	F^Pr1(1)(Y)	F^Pr2(1)(Y)	210.65	225.70	301.93	120.44	126.31	123.83	241.45	243.84	281.75	
F^GK(2)(Y)	F^Pr1(2)(Y)	F^Pr2(2)(Y)	21.64	225.69	301.91	120.45	126.33	123.85	241.45	243.84	281.75	
F^GK(3)(Y)	F^Pr1(3)(Y)	F^Pr2(3)(Y)	210.65	225.70	301.93	120.44	126.31	123.83	241.45	243.84	281.75	
F^GK(4)(Y)	F^Pr1(4)(Y)	F^Pr2(4)(Y)	210.64	225.68	301.89	120.46	126.35	123.87	241.45	243.84	281.75	
F^GK(5)(Y)	F^Pr1(5)(Y)	F^Pr2(5)(Y)	210.92	226.31	301.77	131.42	155.44	151.23	241.47	243.87	281.79	
F^GK(6)(Y)	F^Pr1(6)(Y)	F^Pr2(6)(Y)	210.91	226.30	301.75	131.34	155.16	151.95	241.47	243.87	281.79	
F^GK(7)(Y)	F^Pr1(7)(Y)	F^Pr2(7)(Y)	210.45	225.26	301.34	119.85	125.07	121.61	241.44	243.81	281.72	
F^GK(8)(Y)	F^Pr1(8)(Y)	F^Pr2(8)(Y)	210.92	226.31	301.77	131.42	155.44	151.23	241.47	243.87	281.79	
F^GK(9)(Y)	F^Pr1(9)(Y)	F^Pr2(9)(Y)	210.44	225.26	301.32	119.85	125.07	121.61	241.44	243.81	281.72	
F^GK(10)(Y)	F^Pr1(10)(Y)	F^Pr2(10)(Y)	210.13	224.58	300.42	119.83	125.03	111.57	241.41	243.75	281.65	
	F^(y)			100.00			100.00				100.00	
	F^R(Y)			171.46			44.640				28.330	
	F^P(Y)			29.290			56.480				213.50	
	F^BT,R(Y)			185.21			71.990				241.81	
	F^BT,P(Y)			51.02			88.500				241.41	
	F^Reg(Y)			201.70			101.46				206.53	
	F^R,D(Y)			210.13			119.83				49.610	

Table 15 Percentage relative efficiency using Population 4, 5, 6 when x=Q3(x),y=Q3(y).

Estimators	Population 4	Population 5	Population 6	
F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	F^GK(Y)	F^Pr1(Y)	F^Pr2(Y)	
F^GK(1)(Y)	F^Pr1(1)(Y)	F^Pr2(1)(Y)	245.26	281.27	270.29	110.78	111.13	111.65	228.65	231.74	254.02	
F^GK(2)(Y)	F^Pr1(2)(Y)	F^Pr2(2)(Y)	244.78	280.08	269.15	110.33	111.20	110.72	228.62	231.67	253.94	
F^GK(3)(Y)	F^Pr1(3)(Y)	F^Pr2(3)(Y)	245.76	281.60	271.56	111.23	113.10	111.63	228.69	231.81	254.10	
F^GK(4)(Y)	F^Pr1(4)(Y)	F^Pr2(4)(Y)	244.72	279.94	269.02	110.27	111.09	110.62	228.62	231.66	253.93	
F^GK(5)(Y)	F^Pr1(5)(Y)	F^Pr2(5)(Y)	245.12	280.91	269.95	113.72	119.14	118.62	228.65	231.72	254.00	
F^GK(6)(Y)	F^Pr1(6)(Y)	F^Pr2(6)(Y)	244.91	280.39	269.45	118.02	131.80	131.20	228.63	231.69	253.96	
F^GK(7)(Y)	F^Pr1(7)(Y)	F^Pr2(7)(Y)	245.13	280.93	269.97	110.47	111.50	111.02	228.65	231.72	253.99	
F^GK(8)(Y)	F^Pr1(8)(Y)	F^Pr2(8)(Y)	245.59	281.15	271.13	111.46	115.96	115.46	228.68	231.79	254.07	
F^GK(9)(Y)	F^Pr1(9)(Y)	F^Pr2(9)(Y)	244.74	280.00	269.08	110.25	111.03	11,056	228.62	231.67	253.94	
F^GK(10)(Y)	F^Pr1(10)(Y)	F^Pr2(10)(Y)	244.68	279.68	268.93	110.240	111.02	110.55	228.62	231.66	253.93	
	F^(y)			100.00			100.00				100.00	
	F^R(Y)			213.53			41.450				28.040	
	F^P(Y)			27.780			59.820				199.84	
	F^BT,R(Y)			206.54			68.670				228.42	
	F^BT,P(Y)			49.600			95.800				228.62	
	F^Reg(Y)			241.84			104.44				199.92	
	F^R,D(Y)			244.68			110.24				50.000	

For visualization, we take population 1 and 4 respectively, in descriptions of these graphs we mention that what kind of trash holed we used for finding distribution function. The comparison of numerous estimators in terms of PRE for six populations is depicted in Figs. 1, 2, 3, 4, 5, 6, 7 and 8. The length of a bar is directly associated with the efficiency of an estimator. However, it can be conditional that the suggested estimators, in our case shown by F^Pr1(Y) and F^Pr2(Y), have outperformed the other competitive estimators. Across the suggested class, it is observed that the second proposed class of estimator is more robust than the first proposed class of estimators, because of higher efficiency. Tables 16 and 17 show that the proposed estimators outperform all other estimators currently in use. When X and Y are highly positively correlated, the PRE demonstrates that the second family of estimators proposed in SRS provides a reliable estimate.Figure 1 Percentage of relative efficiencies of existing and proposed estimators when {x=X¯,y=Y¯}, using Population 1.

Figure 2 Percentage of relative efficiencies of existing and proposed estimators when x=Q1(x),y=Q1(y), using Population 1.

Figure 3 Percentage of relative efficiencies of existing and proposed estimators when x=X~,y=Y~, using Population 1.

Figure 4 Percentage of relative efficiencies of existing and proposed estimators when x=Q3(x),y=Q3(y), using Population 1.

Figure 5 Percentage relative efficiencies of existing and proposed estimators when {x=X¯,y=Y¯}, using Population 4.

Figure 6 Percentage of relative efficiencies of existing and proposed estimators when x=Q1(x),y=Q1(y), using Population.

Figure 7 Percentage relative efficiencies of existing and proposed estimators when x=X~,y=Y~, using Population 4.

Figure 8 Percentage relative efficiencies of existing and proposed estimators when x=Q3(x),y=Q3(y), using Population 4.

Table 16 MSEs of population DF estimators using simulation.

Estimator	Population I	Population II	Population III	
F^(y)	0.0022520	0.0022500	0.0022520	
F^R(Y)	0.0064510	0.0026960	0.0012260	
F^P(Y)	0.0026130	0.0064800	0.0077600	
F^BT,R(Y)	0.0037770	0.0018730	0.0011830	
F^BT,P(Y)	0.0018750	0.0038270	0.0044330	
F^Reg(Y)	0.0018580	0.0018530	0.0010650	
F^R,D(Y)	0.0018450	0.0018400	0.0010600	
F^G,K(Y)	0.0018440	0.0018390	0.0010600	
F^Pr1(Y)	0.0018390	0.0018370	0.0010580	
F^Pr2(Y)	0.0017370	0.0016740	0.0008970	

Table 17 PREs of population DF estimators using simulation.

Estimator	Population I	Population II	Population III	
F^(y)	100	100	100	
F^R(Y)	34.90917	83.46483	183.5737	
F^P(Y)	86.18418	34.73246	29.01847	
F^BT,R(Y)	59.62183	120.1159	190.3484	
F^BT,P(Y)	120.0678	58.80547	50.79537	
F^Reg(Y)	121.1700	121.4586	211.4561	
F^R,D(Y)	122.0601	122.3173	212.3715	
F^G,K(Y)	122.0912	122.3487	212.4257	
F^Pr1(Y)	122.4285	122.4696	212.8426	
F^Pr2(Y)	129.6135	134.4366	250.9312	

Conclusion

In this article, we have suggested two improved classes of estimators to estimate the finite population DF using dual auxiliary varaible. The bias and MSE of the suggested classes of estimators are derived up to the first order of approximmation. To observe the efficiency of estimators, six real data sets are used. Also To check the uniqueness and generalizability of the suggested classes of estimaators, we also employ a simulation study. Based on the numerical outcomes, it is observed that the suggested classes of estimators are more efficient than the exisitng estimators, for all the considered populations. The suggested modified classes of estimators F^Pr1(Y) and F^Pr2(Y) perform better as compared to all other considered estimators, although F^Pr2(Y) is the best. The current work can be extended to estimate population mean using calibration approach under stratified random sampling.

Acknowledgements

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R735), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2024/R/1446).

Author contributions

M.S.M. interpretation of the results; wrote the main manuscript S.A. wrote the main manuscript H.M.A. Analysis; wrote the main manuscript F.M.A. Conceptualizations; wrote the main manuscript R.A. helped us to improve the language of the paper; wrote the main manuscript M.E. supervision; wrote the main manuscript S.M.A. helped us in the revised manuscript S.N. helped us to answer all the questions arises during revision.

Data availability

The datasets generated and/or analysed during the current study are available in the current study are available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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